Example 1.5 The trivial sigma algebra is clearly \(\mathcal F_0=\{\emptyset, \Omega\}\) which does not seem very useful.
The partition \(\mathcal P_{equal}=\{\{HH,TT\}, \{HT,TH\}\}\) above, is not a sigma-algebra as it does not contain the empty set. If we add the empty set, then is still not a sigma algebra as it is not closed under union. The union of the only two elements is \(\Omega\). If we include \(\Omega\) then we have the sigma algebra:
\[
\mathcal F_{equal}=\{\emptyset, \Omega, \{HH,TT\}, \{HT,TH\}\}
\]
The partition \(\mathcal P_{heads}\) above is also not a sigma algebra but if we add all possible unions then we obtain the sigma algebra:
\[
\begin{aligned}
\mathcal F_{heads}& =\{\emptyset, \Omega, \{TT\}, \{HT,TH\}, \{HH\}, \{HT,TH,HH\},\\
& \{HT,TH,TT\}, \{HH,TT\}\}
\end{aligned}
\]
The set \[
\mathcal G =\{\emptyset,\Omega,\{HT\},\{TH\},\{HH\},\{TT\}\}
\]
obatined from \(\mathcal P_1\) is neither a partition nor a sigma algebra as it is not closed under union. For example, \(\{HT\}\cup \{TH\}=\{HT,TH\}\notin \mathcal G\). However, if we add all possible unions of the elements of \(\mathcal G\) we obtain the power set of \(\Omega\), that is the set of all subsets of \(\Omega\):
\[
\begin{aligned}
\mathcal F_{max} &= \{\emptyset, \{HT\},\{TH\},\{HH\},\{TT\},\\
& \{HT,TH\}, \{HT,HH\}, \{HT,TT\}, \{TH,HH\}, \{TH,TT\}, \{HH,TT\}\\
& \{HT,TH,HH\}, \{HT,TH,TT\}, \{HT,HH,TT\}, \{TH,HH,TT\},\Omega \}
\end{aligned}
\]
This is the largest possible sigma-algebra for this sample space. It has \(2^4=16\) elements since the sample space has 4 elements. In general, if the sample space has \(n\) elements, then its power set has \(2^n\) elements.
Also generally, if we have a finite partition of \(\Omega\) then the collection of all unions of sets in the partition (including the empty set) is a sigma-algebra.
Note that different sigma algebras serve for different purposes. For example, the sigma algebra \(\mathcal F_{equal}\) is useful if we are only interested in whether the two coin flips are the same or different. The sigma algebra \(\mathcal F_{heads}\) is useful if we areinterested in the number of heads. The power set \(\mathcal F_{max}\) is a sigma algebra that me be more useful if we are interested in all possible events.
In this example we also observe that:
\[
\mathcal F_0 \subset \mathcal F_{equal}
\subset \mathcal F_{heads}
\subset \mathcal F_{max}
\]
so that \(\mathcal F_0\) and \(\mathcal F_{max}\) are the smallest and largest sigma algebras possible for this sample space.